Question: Evaluate the definite integral. $\int^{4\pi}_{3\pi}2\sin(x)\,dx = $
Explanation: First, use the sine rule: $\begin{aligned}\int^{4\pi}_{3\pi}2\sin(x)\,dx~&=~-2\cos(x)\Bigg|^{4\pi}_{{3\pi}}\end{aligned}$ Second, plug in the limits of integration: $(-2\cdot{\cos(4\pi)})-(-2\cdot{\cos(3\pi)}) = -2-2 =-4$. The answer: $\int^{4\pi}_{3\pi}2\sin(x)\,dx~=~-4$